The tangent to a given curve y = f(x) is perpendicular to the x-axis, if

• $\frac{\mathrm{dy}}{\mathrm{dx}} - 1$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 0$

• $\frac{\mathrm{dx}}{\mathrm{dy}} = 1$

• $\frac{\mathrm{dy}}{\mathrm{dx}} = 0$

The minimum value of ${27}^{\cos \left(2\mathrm{x}\right)} {81}^{\sin \left(2\mathrm{x}\right)}$ is

• - 5

• $\frac{1}{5}$

• $\frac{1}{243}$

• $\frac{1}{27}$

A stone is thrown vertically upwards from the top of a tower 64 m high according to the law s = 48t - 16t². The greatest height attained by the stone above ground is

• 36 m

• 32 m

• 100  m

• 64 m

The length of the subtangent at t on the curve $\mathrm{x} = \mathrm{a}\left(\mathrm{t} + \sin \left(\mathrm{t}\right)\right)$, y = $\mathrm{a}\left(1 - \cos \left(\mathrm{t}\right)\right)$

• $\mathrm{asin}\left(\mathrm{t}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)\tan \left(\frac{\mathrm{t}}{2}\right)$

• $2\mathrm{asin}\left(\frac{\mathrm{t}}{2}\right)$

• $2{\mathrm{asin}}^3\left(\frac{\mathrm{t}}{2}\right)\sec \left(\frac{\mathrm{t}}{2}\right)$

685.

A wire of length 20 cm is bent in the form of a sector of a circle. The maximum area that can be enclosed by the wire is

• 20 sq cm

• 25 sq cm

• 10 sq cm

• 30 sq cm

The condition for the line y = mx + c to be a normal to the parabola y² = 4ax is

• c = - 2am - am³

• $\mathrm{c} = - \frac{\mathrm{a}}{\mathrm{m}}$

• $\mathrm{c} = \frac{\mathrm{a}}{\mathrm{m}}$

• c = 2am + am³

P is the point of contact of the tangent form the origin to the curve y = log_e(x). The length of the perpendicular drawn form the origin to the normal at P is

• $\frac{1}{2\mathrm{e}}$

• $\frac{1}{\mathrm{e}}$

• $2\sqrt{{\mathrm{e}}^2 + 1}$

• $\sqrt{{\mathrm{e}}^2 +\hspace{0.17em}1}$

For the curve 4x⁵ = 5y⁴, the ratio of the cube of the subtangent at a point on the curve the square of the subnormal at the same point is

• $\frac{4^4}{5}$

• $\frac{5^4}{4}$

• $\frac{4^4}{5^4}$

• ${\left(\frac{5}{4}\right)}^4$

The set of real values of x for which f(x) = $\frac{\mathrm{x}}{\log \left(\mathrm{x}\right)}$ is increasing is

• $\left\{\mathrm{x} :\hspace{0.17em}\mathrm{x} \ge \mathrm{e}\right\}$

• empty

• {x : x < e}

• {1}

A sphere increases its volume at the rate of $\mathrm{\pi }$ cc/s. The rate at which its surface area increases when the radius is 1 cm is

• $2\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\mathrm{\pi } \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{3\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$

• $\frac{\mathrm{\pi }}{2} \mathrm{sq} \mathrm{cm}/\mathrm{s}$